Related papers: Approximating Ropelength by Energy Functions

We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on…

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we…

Knots across various length scales, from micro to macro-scales, such as polymers, DNA, shoelaces, and surgery, serving their unique mechanical properties. The shape of ideal knots has been extensively studied in the context of knot theory,…

We study the formation of knots on a macroscopic ball-chain, which is shaken on a horizontal plate at 12 times the acceleration of gravity. We find that above a certain critical length, the knotting probability is independent of chain…

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) =…

The statistical mechanics of a long knotted collapsed polymer is determined by a free-energy with a knot-dependent subleading term, which is linked to the length of the shortest polymer that can hold such knot. The only other parameter…

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the…

A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the…

We consider random walk model of a semi-flexible polymer chain on a square and a cubic lattice to enumerate conformations of the polymer chain in two and three dimensions, respectively. The bending energy of the chain is assumed as the key…

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby…

This survey reviews Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and the ribbonlength problem asks to minimize the…

In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its…

Knot and link energies can be computed from sets of closed curves in three dimensional space, and each type of knot or link has a minimum energy associated with it. Here, we consider embeddings of links that locally or globally minimize the…

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots.…

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore…

The mechanical resistance of a polyethylene strand subject to tension and the way its properties are affected by the presence of a knot is studied using first-principles molecular dynamics calculations. The distribution of strain energy for…

The probability of a random polygon (or a ring polymer) having a knot type $K$ should depend on the complexity of the knot $K$. Through computer simulation using knot invariants, we show that the knotting probability decreases exponentially…

We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modelled by a self avoiding polygon of N steps on cubic lattice and l is the…

Probably the most natural energy functional to be considered for knotted strings is the one given by the electrostatic repulsion. In the absence of counter-charges, a charged, knotted string evolving along the energy gradient of…